Question

What is the derivative of `y=(2x+1)^2((x^2+5)/(x^2-2))`?

Answer 1

`(2(2x+1)*(2))*((-14x)/(x^2-2)^2)`

Explanation:

To begin, you want to use the chain rule for the first portion

`(2x+1)^2`

The chain rule is done by finding the derivative of the outside, keeping the inside, and after that you will multiply it by the derivative of the inside.

`f'(g(x)) * g'(x)`

This will give us

`2(2x+1)*(2)`

Next you want to find the derivative of the next function using the quotient rule.

((f'(x)g(x))-(f(x)g'(x)))/(g(x))^2

It will look like

`(((2x)*(x^2-2))-((x^2+5)*(2x))) / (x^2-2)^2`

After you simplify it should look like

`((2x^3-4x)-(2x^3+10x))/(x^2-2)^2`

Then you end up with

`(-14x)/(x^2-2)^2`

Finally you can combine the two to get your final answer

`(2(2x+1)*(2)) * ((-14x)/(x^2-2)^2)`